28 Apr The Quadratic Formula Explained

At Miracle Learning Centre, we teach you interesting stuff in math tuition. In one of our previous maths tuition, we had discussed about quadratic equations. In this math tuition, we are going to further develop our understanding on the topic. If you have not gone through the previous article about quadratic equations, it is highly recommended that you do so before going through this one.

Let us get started with this maths tuition without wasting any further time. Please recall the standard form of the quadratic equations which is:

ax2 + bx + c = 0

and the sridharacharya formula for solving quadratic equations which goes as follows:

x == (-b± √(b^2-4ac))/2a

Converting quadratic equations to the standard form

Some quadratic equations, at first glance, might not look like to represent the standard form which is:

ax2 + bx + c = 0

In this section of this math tuition, we are going to see how to tackle such equations and write them in the standard form.

Consider the following equation:

8- 3×2 = 48x

To write it in the standard form, we will first bring all the terms on one side of the equation. Once we do that, the equation becomes;

8 – 3×2 – 48x = 0

Now, we are going to reorder the terms i.e. we are going to write the term which has x2 as coefficient as the first term, write the term which has x as coefficient as the second term and write the constant part at last. Once we do that, the equation becomes:

-3×2 -48x + 8 =0

Now, we can see that the equation is in the standard form. In this case, a =-3, b=48 and c=8. It is always a good practice to write the quadratic equation in standard form before working with it.

The Discriminant

In the sridharacharya formula, the part = (-b± √(b^2-4ac))/2a is known as the discriminant . From the value of the discriminant, we can predict the properties of the roots of the equation.

If the value of the discriminant is positive, then the equation must have 2 real solutions. The word “real” here implies the fact that the roots can be any number including decimals.

If the value of the discriminant is 0, then the equation has only one real solution. The word “real” has the same implication – the solution is a real number.

If the value of the discriminant is negative, then there are no real solutions to the equation. In this case, there are two complex solutions. Complex solutions means the solutions are complex numbers.

Consider the following equation:

x2 – x – 6 = 0

For the above equation, a = 1, b = -1 and c =-6.

As a result, the discriminant, b^2-4ac = 25 > 0. Since the discriminant is positive, the equation has two solutions.

Indeed, if we plugin the value to the sridharcharya formula, we can see that the solutions are 3 and -2.

Now, consider the following quadratic equation:

x2 – 4x + 4 =0

In this case, a =1, b =-4 and c =4. The discriminant is this case is 0. So we can conclude that the equation has one real solution; the solution is 2 in this case.

For the third scenario, consider the following equation:

x(10-x) -40 =0

Once we simplify, we get the following equation:

10x –x2 – 40 = 0

-x2 + 10x -40 =0

In this case, a =-1, b=10 and c=40. The discriminant is this case is -60 < 0. As a result, this equation has complex numbers as solutions. The solutions in this case are 5+√15i and and 5-√15i where I represents √-1 and it is known as the imaginary part. We already have a tutorial about imaginary numbers and you can use that tutorial as a starting point for understanding complex numbers. So, in this math tuition, we explored quadratic equations in more detail. We also learned about the discriminant and how we can use the value of the discriminant to predict the type of the roots of a quadratic equation. Once you learn about coordinate geometry, you will see how the graph of a quadratic equation looks like. If you have any difficulty in understanding the topics discussed here, feel free to reach out to your math tuition or your peers. A solid understanding of the topics discussed in this article is required in order to do well in math and to understand a lot of other topics which involve quadratic equations. We hope you enjoyed the math tuition from Miracle Learning Centre. Look out for the next math tuition from Miracle Learning Centre.